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184       XIII    INTEGRATION

then so is inf (u, 1) (same method as in (d)). Deduce that, if u and v are pure potentials,
then inf(w, v + 1) is a pure potential, by remarking that v + inf(w, 1) is a pure potential
and using (d). Finally show that, if fe ?% is ^>0 almost everywhere and if ue& is
such that Uf(x) <j H(JC) + 1 almost everywhere w fte set of points xeX such that
f(x) > 0, then \Jf(x) ^ u(x) + 1 almost everywhere in X (" complete maximum prin-
ciple": same method as in (e)).

14. INTEGRATION WITH  RESPECT TO A POSITIVE  MEASURE
WITH BASE   ji

(13.14.1)   Let g be a locally ^.-integrable function which is ^0 on X, and let
v  g  fj.. Then iffZ> 0 -is any R-valued function on X, we have

(13.14.1.1)

f*/Wv= t

J                J

where, on the right-hand side, the value of fg is by definition taken to be 0 at
every point xeXat which one off(x), g(x) vanishes (even if the other factor
is +00 (13.11)).

The proof consists of several steps.

(13.14.1.2)    Suppose first of all that/e</. Then (12.7.8) there exists an in-
creasing sequence (fn) of functions belonging to ?TR(X) such that/= sup/n .

n

In view of the convention about products, this implies that the sequence
(fng) is increasing and that fg is equivalent to sup(/n^), because g is almost

n

everywhere finite (with respect to fj). Moreover, the functions fng are ju-
integrable (13.13.1), and \fngdn=.\fndv. Hence it follows from (13.5.7)
that

/rfv = sup   /Brfv = sup   /B0rf/t=      fgdfi.

(13.14.1.3)    Every ^-negligible set N is also ^-negligible.

Suppose first of all that N is relatively compact. Then ((3.18.2) and
(13.7.9)) there exists a decreasing sequence of relatively compact open
sets Un containing N, such that inf ju(U) = 0. Since gcpl]n is ju-integrable

n

by virtue of (13.13.1),   we   have  J#<pUn dfi = v(Nn)   by   (13.14.1.2)   and Deduce from (b) that an element v e 3f? belongs to & if and only if (v\w) i>0